Download Reduction of order practice problems

Kenyon College
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Math 333
Tuesday, February 26, 2008
Practice Problems: Reduction of Order
For each of the following differential equations, a known solution y1 is given. Use
the method of reduction of order to find a second solution of the given differential
equation. Find the general solution y(t) of the differential equation. Verify that y(t)
is indeed the general solution by showing that y(t) is a linear combination of two
solutions y1 and y2 whose Wronskian is nonzero.
1. t2 y 00 − 4ty 0 + 6y = 0, t > 0; y1 (t) = t2
2. t2 y 00 + 2ty 0 − 2y = 0, t > 0; y1 (t) = t
3. t2 y 00 + 3ty 0 + y = 0, t > 0; y1 (t) = t−1
4. t2 y 00 − t(t + 2)y 0 + (t + 2)y = 0, t > 0, y1 (t) = t
5. xy 00 − y 0 + 4x3 y = 0, x > 0; y1 (x) = sin(x2 )
6. (x − 1)y 00 − xy 0 + y = 0, x > 1; y1 (x) = ex
7. x2 y 00 − (x − 0.1875y) = 0, x > 0; y1 (x) = x1/4 e2
√
x
8. x2 y 00 + xy 0 + (x2 − 0.25)y = 0, x > 0; y1 (x) = x−1/2 sin x
Math 333: Diff Eq
1
Reduction of Order Problems
Kenyon College
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Answers to the reduction of order practice problems.
1. y2 (t) = t3
2. y2 (t) = t−2
3. y2 (t) = t−1 ln t
4. y2 (t) = tet
5. y2 (x) = cos(x2 )
6. y2 (x) = x
7. y2 (x) = x1/4 e−2
√
x
8. y2 (x) = x−1/2 cos x
Math 333: Diff Eq
2
Reduction of Order Problems